Vectors
A line segment to which a direction has been assigned is called a directed line segment. The figure below shows a directed line segment form Pto Q. We call
Pthe initial point and Qthe terminal point. We denote this directed line segment by PQ.
The magnitude of the directed line segment PQis its length. We denote this by || PQ ||. Thus, || PQ || is the distance from point Pto point Q. Because distance is nonnegative, vectors do not have negative magnitudes. Geometrically, a vector is a directed line segment. Vectors are often denoted by a boldface letter, such as v. If a vector vhas the same magnitude and the same direction as the directed line segment PQ, we write
v= PQ.
P
Q
Initial point
Terminal point
Directed Line Segments and
Geometric Vectors
Vector Multiplication
If
k
is a real number and
v
a vector, the vector
k
v
is
called a
scalar multiple
of the vector v. The
magnitude and direction of
k
v
are given as
follows:
The vector
k
v
has a magnitude of |
k
| ||
v
||. We
describe this as the absolute value of
k
times the
magnitude of vector
v
.
The vector
k
v
has a direction that is:
the same as the direction of
v
if
k
> 0, and
A geometric method for adding two vectors is shown below. The sum of u+ v
is called the resultant vector. Here is how we find this vector. 1. Position uand vso the terminal point of uextends from the initial
point of v.
2. The resultant vector, u+ v, extends from the initial point of uto the terminal point of v. Initial point of u u + v v u Resultant vector Terminal point of v
The Geometric Method for
Adding Two Vectors
The difference of two vectors, v–u, is defined as v–u= v+ (-u), where –uis the scalar multiplication of uand –1: -1u. The difference v–uis shown below geometrically. v u -u -u v – u
The Geometric Method for the
Difference of Two Vectors
1 1 i j O x y
The i and j Unit Vectors
• Vector
i
is the unit vector whose direction is along
the positive
x
-axis. Vector
j
is the unit vector
whose direction is along the positive
y
-axis.
Representing Vectors in
Rectangular Coordinates
Vector
v
, from (0, 0) to (a, b), is represented as
v
=
a
i
+
b
j
.
The real numbers a and b are called the scalar
components of
v
. Note that
a
is the horizontal component of
v
, and
b
is the vertical component of
v
.
The vector sum
a
i
+
b
j
is called a linear combination
of the vectors
i
and
j
. The magnitude of
v
=
a
i
+
b
j
is given by
v
=
a
2+
b
2Sketch the vector v= -3i+ 4jand find its magnitude.
Solution For the given vector v= -3i+ 4j, a = -3 and b = 4. The vector, shown below, has the origin, (0, 0), for its initial point and (a, b) = (-3, 4) for its terminal point. We sketch the vector by drawing an arrow from (0, 0) to (-3, 4). We determine the magnitude of the vector by using the distance formula. Thus, the magnitude is
-5-4 -3 -2-1 12345 5 4 3 2 1 -1 -2 -3 -4 -5 Initial point Terminal point v = -3i + 4j v= a2+b2 = (−3)2 +42 = 9+16 = 25=5
Text Example
Representing Vectors in
Rectangular Coordinates
• Vector
v
with initial point
P
1= (
x
1,
y
1) and
terminal point
P
2= (
x
2,
y
2) is equal to the position
vector
Adding and Subtracting Vectors
in Terms of i and j
• If v
=
a
1i
+
b
1j
and w
=
a
2i
+
b
2j, then
•
v
+ w
= (
a
1+
a
2)i
+ (
b
1+
b
2)j
•
v
–
w
= (
a
1–
a
2)i
+ (
b
1–
b
2)j
If v= 5i+ 4jand w= 6i– 9j, find: a. v+ w b. v–w. Solution• v+ w= (5i+ 4j) + (6i– 9j) These are the given vectors. = (5 + 6)i+ [4 + (-9)]j Add the horizontal components. Add the
vertical components. = 11i– 5j Simplify.
• v+ w= (5i+ 4j) – (6i– 9j) These are the given vectors.
= (5 – 6)i+ [4 – (-9)]j Subtract the horizontal components. Subtract the vertical components.
= -i+ 13j Simplify.
